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On the different definitions of the stability of functional equations. (Sur les définitions différentes de la stabilité des équations fonctionnelles.) (French) Zbl 1060.39031
In a great number of papers and monographs dealing with the stability of functional equations, various types of this concept are considered. The paper under review is a kind of survey; the main types of the stability are defined and compared. Here are the list of them: Let $(*)$ $L(f)=R(f)$ be a functional equation with an unknown function $f$ and let $\rho$ be a metric in the target space. The equation $(*)$ is (uniquely) stable if for each $\varepsilon>0$ there exists $\delta>0$ such that for each $g$ satisfying $\rho(L(g),R(g))<\delta$, for all the variables of $g$, there exists (a unique) solution $f$ of the equation such that $\rho(g,f)<\varepsilon$ (for all the variables of $f$ and $g$). The equation $(*)$ is (uniquely) $b$-stable if for each $g$ for which $\rho(L(g),R(g))$ is bounded, there exists (a unique) solution $f$ of $(*)$ such that $\rho(f,g)$ is bounded. One says that $(*)$ is (uniquely) uniformly $b$-stable if for each $\delta>0$ there exists $\varepsilon>0$ such that for each $g$ satisfying $\rho(L(g),R(g))<\delta$ there is a (unique) solution $f$ of $(*)$ such that $\rho(f,g)<\varepsilon$. In particular, if for some $\alpha>0$, $\varepsilon=\alpha\delta$, $(*)$ is said to be strongly stable (or strongly and uniquely stable). There are also considered definitions of not uniquely and totally not uniquely stability as well as (uniquely/not uniquely/totally not uniquely) iterative stability. The equation $(*)$ is superstable if for each $g$ for which $\rho(L(g),R(g))$ is bounded, $g$ is bounded or it is a solution of the equation $(*)$; if functions $g$, in the case considered, are bounded by the same constant, $(*)$ is called strongly superstable. If for each $g$ for which $\rho(L(g),R(g))$ is bounded, $g$ a solution of $(*)$, then we call $(*)$ completely superstable. There are suitable examples for the above definitions and comparisons. Also some properties and related results are proved. The so-called Hyers’ operator and the stability of conditional functional equations are also mentioned.

39B82Stability, separation, extension, and related topics
39-02Research monographs (functional equations)
39B52Functional equations for functions with more general domains and/or ranges
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